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Isaac Newton's method is perhaps the best known method for finding successively better
approximations to the zeroes of a
real
valued
function
. Newton's method can often converge
remarkably quickly; especially if the iteration begins "sufficiently near" the desired root. Just
how near "sufficiently near" needs to be, and just how quickly "remarkably quickly" can be,
depends on the problem. This is discussed in detail below. Unfortunately, when iteration begins
far from the desired root, Newton's method can easily lead an unwary user astray with little
warning. Thus, good implementations of the method embed it in a routine that also detects and
perhaps overcomes possible convergence failures.
Given a function
ƒ
(
x
) and its
derivative
ƒ
'(
x
), we begin with a first guess
x
0
. A better
approximation
x
1
is
Using the derivate definition to find an equation of tangent line to the curve Y = X^3 + 2X at the
point (1, 3).
1.
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 Fall '09
 moi
 Approximation

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